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Coend calculus

Guest post by Fosco Loregian and Bryce Clarke.

Pastro, Craig, and Ross Street. Doubles for monoidal categories. Theory and applications of categories 21.4 (2008): 61-75.

Coend calculus rules the behaviour of certain universal objects associated to functors of two variables T:C op×CDT : C^{op}\times C \to D; intuitively, end(T)end(T) stands to TT as the limit limF\lim\, F of F:ABF : A \to B stands to FF; the major difference is that end(T)end(T) takes into account the fact that TT eats at the same time two “terms” of the same “type” CC, once covariantly in the second component, and once contravariantly in the first: on arrows f:ccf : c\to c' the functor TT acts in fact as follows:

T(c,c) T(f,c) T(c,f) T(c,c) T(c,c) \begin{array}{ccccc} && T(c',c) && \\ &\overset{T(f,c)}\swarrow && \overset{T(c',f)}\searrow &\\ T(c,c) &&&& T(c',c') \end{array}

Now, a distinguising feature of the objects that depend contra-covariantly on the same variable is that they can be integrated: given a sufficiently regular function f(x)f(x), its dependence from xx can be thought as “covariant” (and defined, say, on a topological vector space VR nV\cong \mathbf{R}^n), whereas the “dxd x” in the symbol

f(x)dx \int f(x)d x

is contravariant (it belongs to a certain dual space of covectors on VV); altogether, the integral can be thought as exhibiting a contra-covariant dependence from xx.

Ends and coends are associated to functors T:C op×CDT : C^{op}\times C \to D in a similar fashion that resembles integration: they are certain objects cT(c,c)\int_c T(c,c) (the end) and cT(c,c)\int^c T(c,c) (the coend), canonically associated to TT, treating cc as a mute variable (meaning that cT(c,c)\int_c T(c,c) and cT(c,c)\int_{c'} T(c',c') are the same object), and satisfying a “commutative rule of integrals” analogous to

f(x,y)dxdy=(f(x,y)dx)dy=(f(x,y)dy)dx \int f(x,y) d x d y = \int \Big(\int f(x,y)d x\Big) d y = \int \Big(\int f(x,y)d y\Big) d x

The end of TT is endowed with projections on the “symmetrized components” cT(c,c)T(c,c)\int_c T(c,c) \to T(c',c'), one for each object cc'; dually, the coend cT(c,c)\int^c T(c,c) is endowed with injections T(c,c) cT(c,c)T(c',c') \to \int^c T(c,c).

All in all, this happens also for colimits, so the two constructions are -at least intuitively- tightly related: this intuition can of course be made more precise. Every universal object built in category theory can be thought either as a subobject of a product (a limit), or as a quotient of a coproduct (a colimit), and co/ends make no exception:

  • An end arises as an “object of invariants” cT\int_c T for the action of TT given by the functions on arrows T xy:hom C op×C(x,y)hom D(Tx,Ty)T_{x y} : \hom_{C^{op}\times C}(x,y) \to \hom_D(T x, T y), and it is defined as the subobject of cCT(c,c)\prod_{c\in C} T(c,c) of those elements invariant under this action.
  • Dually, a coend arises as a “quotient space” of cCT(c,c)\coprod_{c\in C}T(c,c) by a suitable equivalence relation generated by the same functions T xy:hom C op×C(x,y)hom D(Tx,Ty)T_{x y} : \hom_{C^{op}\times C}(x,y) \to \hom_D(T x, T y), i.e. as a space of orbits for the action.

More on this will be explained later on in the discussion.

What is more important now, and quite astounding, is that such contra-covariant actions arise at every corner of category theory: using co/ends it is possible to re-state the Yoneda lemma and the theory of Kan extensions, and to find plenty of applications to algebra, topology, geometry… and functional programming. :-)

Dinaturality

As already said, a functor T:C op×CDT : C^{op}\times C \to D acts on morphisms as

T(c,c) T(f,c) T(c,f) T(c,c) T(c,c) \begin{array}{ccccc} && T(c',c) && \\ &\overset{T(f,c)}\swarrow && \overset{T(c',f)}\searrow &\\ T(c,c) &&&& T(c',c') \end{array}

given two such functors, say P,Q:C op×CDP,Q : C^{op}\times C \to D, we can consider the two diagrams

P(c,c) P(f,c) P(c,f) P(c,c) P(c,c) \begin{array}{ccccc} && P(c',c) && \\ &\overset{P(f,c)}\swarrow && \overset{P(c',f)}\searrow &\\ P(c,c) &&&& P(c',c') \end{array}

and

Q(c,c) Q(c,c) Q(c,f) Q(f,c) Q(c,c) \begin{array}{ccccc} Q(c,c)&&&& Q(c',c')\\ &\underset{Q(c,f)}\searrow&& \underset{Q(f,c')}\swarrow& \\ &&Q(c,c')&& \end{array}

Given two functors F,G:ABF,G : A \to B a natural transformation can be seen as a family of maps that “fill the gap” between F(f)F(f) and G(f)G(f) in a commutative square; in a similar fashion, a dinatural transformation between the above P,QP,Q can be seen as a way to close a certain diagram that testifies a transformation from the arrow action of PP to the arrow action of QQ:

P(c,c) P(f,c) P(c,f) P(c,c) P(c,c) α c α c Q(c,c) Q(c,c) Q(c,f) Q(f,c) Q(c,c) \begin{array}{ccccc} && P(c',c) && \\ &\overset{P(f,c)}\swarrow && \overset{P(c',f)}\searrow &\\ P(c,c) &&&& P(c',c')\\ \!\!\!\!{\color{red} \alpha_c\downarrow} &&&&\,\,\,\,\,\,\,{\color{red}\downarrow\alpha_{c'}} \\ Q(c,c)&&&& Q(c',c')\\ &\underset{Q(c,f)}\searrow&& \underset{Q(f,c')}\swarrow& \\ &&Q(c,c')&& \end{array}

Just as co/limits are defined via suitable transformation to/from a constant, so are co/ends:

  • If Q:C op×CDQ : C^{op}\times C \to D, a wedge for QQ with base xx consists of a dinatural transformation from the constant functor on xDx\in D, i.e. of a family of morphisms α c:xQ(c,c)\alpha_c : x\to Q(c,c) such that for each f:ccf :c\to c' the above hexagon reduces to a commutative square:
    x Q(c,c) Q(c,c) Q(c,c) \begin{array}{ccc} x &\to& Q(c,c)\\ \downarrow && \downarrow\\ Q(c',c') &\to & Q(c,c') \end{array}
  • If P:C op×CDP : C^{op}\times C \to D, a cowedge for PP with tip yy consists of a dinatural transformation to the constant functor on yDy\in D, i.e. of a family of morphisms α c:P(c,c)y\alpha_c : P(c,c) \to y such that for each f:ccf :c\to c' the above hexagon reduces to a commutative square:
    P(c,c) P(c,c) P(c,c) y \begin{array}{ccc} P(c',c) &\to& P(c,c)\\ \downarrow && \downarrow\\ P(c',c') &\to & y \end{array}

    There exists a category Wd(Q)Wd(Q) of wedges, defined with an obvious choice of morphisms between bases; similarly, there is a category of cowedges Cwd(P)Cwd(P).

The end cQ(c,c)\int_c Q(c,c) of QQ is now defined as a terminal object in the category of its wedges; dually, the coend cP(c,c)\int^c P(c,c) of PP is the initial object of the category of its cowedges. Of course, we say “the” end because such an initial object is unique up to unique isomorphism when it exists.

So far, so good. In fact, we didn’t stray much far from plain old category theory, as it is possible to show the following:

Lemma (co/ends are colimits): Given Q:C op×CDQ : C^{op}\times C \to D there exist a category twC\text{tw}\, C and a functor Q τ:twCDQ^\tau : \text{tw}\, C \to D such that

cQ(c,c)limQ τ \int_c Q(c,c) \cong \lim \,Q^\tau

For those who know: the end of QQ is the weighted colimit of Q:C op×CDQ : C^{op}\times C \to D with the hom C\hom_C functor C op×CSetC^{op}\times C \to Set, and thus the category twC\text{tw}\, C is nothing more, nothing less than the category of elements of hom C\hom_C; this allows for a very explicit presentation of twC\text{tw}\, C:

  • objects: the arrows of CC, f:ccf : c \to c';
  • morphisms: the commutative squares
    c d f g c d \begin{array}{ccc} c &\leftarrow& d \\ {}^f\downarrow && \downarrow^g\\ c' &\to& d' \end{array}

Corollary (hom commutes with all ends): As a consequence of the fact that cQ\int_c Q is a limit, there is an isomorphism

hom C(y, cP(c,c)) chom C(y,P(c,c))(ccnt) \hom_C\Big(y, \int_c P(c,c)\Big) \cong \int_c \hom_C(y, P(c,c)) \qquad\qquad{(ccnt)}

natural in yy. Dually,

hom C( cP(c,c),y) chom C(P(c,c),y).(ccnt) \hom_C\Big( \int^c P(c,c), y \Big )\cong \int_c \hom_C(P(c,c), y). \qquad\qquad{(ccnt)}

natural in yy.

(of course, a coend is just an end in the opposite category!)

But why are co/ends denoted as integrals? The notation dates back to Yoneda,

Yoneda, Nobuo. “On Ext and exact sequences.” J. Fac. Sci. Univ. Tokyo Sect. I 8.507-576 (1960): 1960.

(in particular, see §4 but beware that the notation is reversed; a coend is denoted a\int_a and an end a *\int_a^\ast!) and it is essentially motivated by the fact that an end behaves like an integral:

Theorem (Fubini rule): Let P:C op×C×D op×DEP : C^{op}\times C \times D^{op}\times D \to E be a functor; then

c( dP(c,c,d,d)) (c,d)P(c,d,c,d) d( cP(c,c,d,d))(Fub) \int^c\left(\int^d P(c,c,d,d)\right) \cong \int^{(c,d)}P(c,d,c,d) \cong \int^d\left(\int^c P(c,c,d,d)\right) \qquad\qquad{(Fub)}

in the sense that if one of the three objects exists, so do the other two, and they are all canonically isomorphic (the category C op×C×D op×DC^{op}\times C \times D^\text{op}\times D is of course equal to (C×D) op×(C×D)(C\times D)^\text{op}\times( C \times D)). Similarly, there is such a rule for ends.

Thus, in category theory integration with respect to a variable is a process that can happen in whatever order we desire: given a permutation σ\sigma of the set {1,,n}\{1,\dots,n\}, whenever the integral

c σ1 c σ2 c σnP(c σ1,c σ1,c σ2,c σ2,,c σn,c σn) \int^{c_{\sigma 1}}\int^{c_{\sigma 2}}\cdots \int^{c_{\sigma n}} P(c_{\sigma 1}, c_{\sigma 1}, c_{\sigma 2}, c_{\sigma 2}, \dots, c_{\sigma n}, c_{\sigma n})

exists, then so does

(c 1,,c n)P(c 1,c 1,c 2,c 2,,c n,c n) \int^{(c_1,\dots, c_n)} P(c_1, c_1, c_2, c_2, \dots, c_n, c_n)

Proof. The slickest proof I know for this goes as follows: assume all coends exist; then, sending a functor P:C op×CDP : C^{op}\times C \to D to its coend is a functor C:[C op×C,D]D\int^C : [C^{op}\times C, D] \to D, and it is easy to see that it is a left adjoint (for those who know, C\int^C is a particular kind of weighted colimit, and every such weighted colimit admits a right adjoint expressed in terms of the weight: but now it’s easy to prove that these right adjoints commute, thus yielding the Fubini rule by uniqueness of adjoints).

The building blocks of co/end calculus

Here we explore how co/ends allow to rediscover category theory from scratch.

Natural transformations

Theorem (Natural transformations as an end) Let F,G:CDF,G : C \to D be two functors; then, there is an isomorphism

Nat(F,G) chom D(Fc,Gc)(nat) Nat(F,G) \cong \int_c \hom_D(F c,G c) \qquad\qquad{(nat)}

Proof. There is a natural choice for a wedge ω:Nat(F,G)hom(Fc,Gc)\omega : Nat(F,G) \to \hom(F c,G c), that sends a natural transformation to its cc-component; it remains to show that this is indeed a terminal wedge. Given another wedge α:Ahom(Fc,Gc)\alpha : A \to \hom(F c, G c), the wedge condition translates into the equation

A α ac hom(Fc,Gc) α ac Gf hom(Fc,Gc) Ff hom(Fc,Gc) \begin{array}{ccc} A &\overset{\alpha_{a c}}\to& \hom(F c,G c) \\ {}_{\alpha_{a c'}}\downarrow && \downarrow_{G f \circ -} \\ \hom(F c', G c') &\underset{- \circ F f}\to & \hom(F c, G c') \end{array}

valid for every aAa\in A; but this is only a convoluted way to say that for every aAa\in A the family

{α a,c:FcGc} \{\alpha_{a,c} : F c \to G c\}

is a natural transformation.

Two important remarks:

  1. In an additive setting, the wedge condition for α\alpha can be easily translated into the fact that natural transformations appear form the kernel of a certain map; the intuition that naturality is a cocycle condition is more or less what led Yoneda to study ends and coends in homological algebra.

  2. Even in a non-additive setting, one can easily see that a natural transformation α:FG\alpha : F \Rightarrow G is a map that equalizes the action of F,GF,G on arrows; this means that the following diagram

    Nat(F,G) cChom(Fc,Gc)Ff *Gf * cchom(Fc,Gc) Nat(F,G) \to \prod_{c\in C} \hom(F c, G c) \underset{G f_\ast}{\overset{F f^\ast}\rightrightarrows} \prod_{c\to c'} \hom(F c, G c')

    is an equalizer; there is nothing special here, as for every functor T:C op×CDT : C^{op}\times C \to D there is a similar equalizer diagram

    cT(c,c) cCT(c,c)Tf *Tf * ccT(c,c) \int_c T(c,c) \to \prod_{c\in C} T(c,c) \underset{T f_\ast}{\overset{T f^\ast}\rightrightarrows} \prod_{c\to c'} T(c, c')

Here‘s a discussion on what is the coend of the hom functor; I claim that the following object represents the coend of hom(F,G)\hom(F-,G-), perhaps you know where the same object appears under a different name, and where it is used for some purpose? I find this particularly intriguing in the case of a monoid MM regarded as single-object category:

  • the end of hom M\hom_M is the center of the monoid, i.e. the subset {mMmx=xmxM}\{m\in M\mid mx=xm \forall x\in M\};
  • the coend of hom M\hom_M corresponds to something like the π 0\pi_0 of the monoid.

I didn’t expect these construction to be dual, and yet they are!

Claim (but also: exercise for the reader). Let CC be a small category. The coend

chom C(c,c) \int^c \hom_C(c,c)

is the set of connected components of the “endo-comma” category whose objects are endomorphisms of CC, and whose morphisms (u:xx)(v:yy)(u : x \to x) \to (v : y \to y) are the f:xyf : x \to y such that fu=vff u = v f. More generally, if F,G:CDF,G: C \to D are functors there is an isomorphism

chom(Fc,Gc)π 0((F/G) end) \int^c \hom(F c, G c) \cong \pi_0((F/G)_{end})

where the endomorphism comma is defined similarly.

The Yoneda lemma and Kan extensions

On the first day He created the Yoneda lemma, and He saw that it was good:

Theorem (The ninja Yoneda lemma) Let F:C opSetF : C^{op} \to Set; then for every object aCa\in C,

Fa cFc×hom C(a,c) cSet(hom C(c,a),Fc) F a\cong \int^c F c \times \hom_C(a,c) \cong \int_c \Set(\hom_C(c,a), F c)

Proof. The set cSet(hom C(c,a),Fc)\int_c \Set(\hom_C(c,a), F c) is the set of natural transformations from hom(,c)\hom(-,c) to FF, and thus the non-ninja Yoneda lemma yields an isomorphism between this set and FaF a. Dually, call FyF\otimes y the functor

a xFx×hom C(a,x) a\mapsto \int^x F x\times \hom_C(a,x)

Then, we have

Nat(Fy,G) aSet(Fx×hom C(a,x),Ga) a xSet(Fx×hom C(a,x),Ga) a xSet(Fx,[hom C(a,x),Ga]) xSet(Fx, a[hom C(a,x),Ga]) xSet(Fx,Nat(y(x),G)) Nat(F,G) \begin{array}{rl} Nat(F\otimes y, G) &\cong \int_a Set(F x\times \hom_C(a,x), G a)\\ &\displaystyle\cong\int_a\int_x Set(F x\times \hom_C(a,x), G a)\\ &\displaystyle\cong\int_a\int_x Set(F x, [\hom_C(a,x), G a])\\ &\displaystyle\cong\int_x Set(F x, \int_a[\hom_C(a,x), G a])\\ &\displaystyle\cong\int_x Set(F x, Nat(y(x), G))\\ &\displaystyle\cong Nat(F, G) \end{array}

Each of these steps can be easily justified in light of what we already proved:

  • the fact that the hom functor preserves ends;
  • the Fubini rule for ends;
  • the fact that the set of natural transformations between two functors is an end.

This natural deduction-style kind of proof is half-jokingly called “coend-fu” ((端楔術 : literally “the art [of handling] terminal wedges”) in my note, soon-ish a book, on coends.

Incidentally, the isomorphism Fa cFc×hom C(a,c)F a\cong \int^c F c \times \hom_C(a,c) is precisely the sense in which “every presheaf is a colimit of representable functors”; the colimiting diagram has domain the category of elements of FF, and the natural functor Σ:El(F)C[C op,Set]\Sigma : El(F) \to C \to [C^{op},Set] has colimit FF.

On the second day, He created Kan extensions, and category theory was complete.

Theorem (Kan extensions are co/ends) Let AGBFCA \xleftarrow{G} B\xrightarrow{F} C be a span of functors; if the coend

bhom A(Gb,a)×Fb \int^b \hom_A(G b, a) \times F b

exists, then it is the value at aa of the left Kan extension of FF along GG; dually, is the end

bFb hom A(a,Gb) \int_b F b ^ {\hom_A(a, G b)}

exists, then it is the value at aa of the right Kan extension of FF along GG.

Proof. The proof is another lengthy, but completely straightforward, kata using the coend-fu we already know:

Nat(Lan GF,H) ahom(Lan GF(a),Ha) ahom( chom(Gc,a)×Fc,Ha) Fub achom(hom(Gc,a)×Fc,Ha) achom(Fc,[hom(Gc,a),Ha]) ccnt chom(Fc, a[hom(Gc,a),Ha]) nat chom(Fc,Nat(hom(Gc,),H)) nat chom(FC,HGC)=Nat(F,HG) \begin{array}{rl} Nat(\text{Lan}_G F, H) &\textstyle \cong \int_a \hom(\text{Lan}_G F( a ), H a ) \\ & \textstyle \cong \int_a \hom\Big( \int^c \hom(G c, a )\times F c, H a \Big) \\ Fub & \textstyle \cong \int_{ a c} \hom\Big( \hom(G c, a )\times F c, H a \Big) \\ & \textstyle \cong \int_{ a c} \hom\Big( F c,[\hom(G c, a ), H a ] \Big) \\ ccnt & \textstyle \cong \int_c \hom\Big( F c,\int_a [\hom(G c, a ), H a ] \Big) \\ nat & \textstyle \cong \int_c \hom\Big( F c,Nat(\hom(G c,-), H) \Big) \\ nat & \textstyle \cong \int_c \hom\Big( F C,H G C \Big) = Nat(F, H G) \\ \end{array}

Bimodules

This Pastro and Street paper makes heavy use of the theory of bimodules. Let’s dig deeper into their structure. First of all, let us define a bicategory as follows:

  • objects are (unitary) rings R,S,R,S,\dots
  • 1-cells M:RSM : R \to S are modules SM R{}_S M_R, left over RR and right over SS.
  • 2-cells φ: RM S RN S\varphi : {}_R M_S \to {}_R N_S are RR-SS-linear homomorphisms of modules.

The composition of 1-cells is the tensor product of modules: RM SN T{}_R M \otimes_S N_T is a RR-TT-bimodule for every RR-SSbimodule MM and every SS-TT-bimodule NN (so in particular this is not a strict 2-category); 2-cells are composed horizontally and vertically, using the obvious function composition, and bifunctoriality of the \otimes operation.

Good! There is a 2-categorical characterization of modules. What is good for? Well: rings are monoids in the category of abelian groups. We could have done the very same thing taking (plain) monoids, i.e. monoids in the category of sets, and defining a category of “bimodules” as sets with a left action of some monoid, and a right action of some other monoid.

What is interesting now is that we can generalize the same construction, with no additional cognitive loading, and take X,Y,ZX,Y,Z\dots to be multi-object monoids: we define a bicategory ModMod as follows:

  • objects are categories C,D,E,C,D, E,\dots
  • 1-cells P:CDP : C \mathrel{⇸} D are functors P:D op×CSetP : D^{op}\times C \to Set
  • 2-cells are natural transformations between functors.

A functor P:D op×CSetP : D^{op}\times C \to Set is a multiobject module on which the multiobject monoids C,DC,D act.

So categories really are monoids, and are also eager to act on sets.

A bimodule is also called a profunctor, a distributor, a correspondence, a span,… and possibly with many other names; each of these names comes from a certain intuition behind their nature that leads to the definition of the same bicategory:

  • they are called profunctors because they generalize functors: some profunctors are called representable, and they are the ones of the form hom(b,Fa)\hom(b, F a) for some functor F:ABF : A \to B between categories. A pro-functor thus works “on behalf of a functor”, as well as a relation generalizes a function.

  • …which is why some people prefer to call them relators: just as a func-tion is a special kind of rela-tion, a func-tor is a special kind of rela-tor.

  • they are called distributors: as the nLab says,

    > Jean Bénabou, who invented the term and originally used “profunctor,” now prefers “distributor,” which is supposed to carry the intuition that a distributor generalizes a functor in a similar way to how a distribution generalizes a function.

    There’s in fact a beautiful story about this: Lawvere defined a notion of distribution between toposes, such that the points of a topos p:Setp : Set \to \mathcal{E} behave like Dirac delta functions, and such that distributions between presheaf toposes are exactly profunctors.

  • they are called correspondences or spans because of the Grothendieck construction: every presheaf P:D op×CSetP : D^{op}\times C \to Set has a category of elements El(P)El(P) that in this case is a discrete fibration over D op×CD^{op}\times C

Well… until now we cheated a bit. In order to get a bicategory we need to define a composition law for 1-cells and show that it is bifunctorial, and I didn’t tell you how to do it: but it turns out it is really easy, if you know coend-fu! Indeed, the intuition of a bimodule as a “matrix indexed by its domain” and the rule to compose two relations guide us to find an expression to meaningfully compose Q:AB,P:BCQ : A \mathrel{⇸} B, P : B \mathrel{⇸} C. We can define

(PQ)(a,c)= bP(a,b)×Q(b,c) (P \diamond Q) (a,c) = \int^b P(a,b)\times Q(b,c)

boils down, on discrete domains A,BSetA,B\in\Set to a “matrix product of sets” like

(PQ) ac= bBP ab×Q bc (P \diamond Q)_{a c} = \sum_{b\in B} P_{a b}\times Q_{b c}

There is also a connection between the ways in which profunctors compose, and the way in which relations do. Indeed, look how the two concepts closely resemble each other:

(x,z)SR yY ((x,y)R) ((y,z)S) (PQ)(x,z) = y Q(x,y) × P(y,z) \begin{array}{cccccc} (x,z)\in S\circ R & \iff & \exists y\in Y & \big((x,y)\in R\big) & \wedge & \big((y,z)\in S\big) \\ (P\diamond Q)(x,z) & = & \int^y & Q(x,y) & \times & P(y,z) \\ \end{array}

Finally: yes, you can make this precise by saying that sets are discrete categories, or even more precisely categories enriched over truth values, and that relations are precisely the {0,1}\{0,1\}-enriched version of profunctors, as they are functions A×B{0,1}A\times B \to \{0,1\}!

Doubles for monoidal categories

One of the key results of the paper Doubles for Monoidal Categories is the canonical equivalence of categories:

Tamb(𝒞)[Doub(𝒞),Set] \mathbf{Tamb}(\mathcal{C}) \simeq [\mathbf{Doub}(\mathcal{C}), \mathbf{Set}]

This theorem has since been labelled by some as the “fundamental theorem of optics”, as it provides the link between the category of Tambara modules Tamb(𝒞)\mathbf{Tamb}(\mathcal{C}) and the double Doub(𝒞)\mathbf{Doub}(\mathcal{C}) of the monoidal category 𝒞\mathcal{C}, now also known as the category of optics. To unpack this theorem, we first begin with the definition of a Tambara module.

Tambara modules

The category of (bi)modules Mod\mathbf{Mod} forms a bicategory, however when we choose a particular category 𝒞\mathcal{C} we may instead consider the monoidal category Mod(𝒞)\mathbf{Mod}(\mathcal{C}) whose:

  • objects are endomodules P:𝒞 op×𝒞SetP \colon \mathcal{C}^{op} \times \mathcal{C} \rightarrow \mathbf{Set};
  • morphisms are natural transformations;
  • monoidal product is module composition:
    (PQ)(X,Y)= ZP(X,Y)×Q(Y,Z) (P \diamond Q) (X,Y) = \int^Z P(X,Y)\times Q(Y,Z)

    One may ask the question: what happens when 𝒞\mathcal{C} has the structure of a monoidal category?

Definition: Let 𝒞\mathcal{C} be a monoidal category. A (left) Tambara module on 𝒞\mathcal{C} consists of:

  • a profunctor P:𝒞 op×𝒞SetP \colon \mathcal{C}^{op} \times \mathcal{C} \rightarrow \mathbf{Set};
  • a family of functions τ A(X,Y):P(X,Y)P(AX,AY)\tau_{A}(X,Y) \colon P(X, Y) \rightarrow P(A \otimes X, A \otimes Y) called the Tambara structure maps, which are natural in X,YX, Y and dinatural in AA, satisfying the equations:

Last revised on August 19, 2019 at 08:45:31. See the history of this page for a list of all contributions to it.